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Abstract

OFDM suffers from inter-carrier interference (ICI) when the channel is time varying.
This article seeks to quantify the amount of interference resulting from wideband
OFDM channels, which are assumed to follow the multi-scale multi-lag (MSML) model.
The MSML channel model results in full channel matrices both in the frequency and
time domains. However, banded approximations are possible, leading to a significant
reduction in the equalization complexity. Measures for determining whether a time-domain
or frequency-domain approach should be undertaken are provided based on the interference
analysis, and we propose to use the conjugate gradient (CG) algorithm to equalize
the channel iteratively. The suitability of a preconditioning technique, that often
accompanies the CG method to accelerate the convergence, is also discussed. We show
that in order for the diagonal preconditioner to function properly, optimal resampling
is indispensable.

Introduction

With many desirable properties such as a high spectral efficiency and inherent resilience
to the multipath dispersions of frequency-selective channels, the orthogonal frequency
division multiplexing (OFDM) technology shows attractive features to wireless radio
applications
[1]. OFDM relies on the assumption that the channel stays constant within at least one
OFDM symbol period. When Doppler effects due to temporal channel variation cannot
be ignored, this assumption does not hold any more since the communication channel
becomes time varying. The Doppler effects corrupt the orthogonality among OFDM subcarriers
by inducing non-negligible inter-carrier interference (ICI)
[2], and can therefore severely deteriorate the system performance. For traditional terrestrial
radio systems, compensation of ICI in channel equalization has widely been researched
for narrowband systems. Due to the small relative signal bandwidth (actual bandwidth divided by
the center frequency) of narrowband systems, the Doppler effects can be modeled primarily
by frequency shifts
[3,4], in which case it is reasonable to assume that each OFDM subcarrier experiences a
statistically identical frequency offset
[2]. Consequently, the effective channel matrix of a narrowband OFDM system in the presence
of Doppler can be approximated as banded. Efficient equalization schemes for such
a banded channel matrix have been studied in, e.g.,
[5-7].

In a wideband system, where the relative signal bandwidth is large, the Doppler effects should
be more appropriately modeled as scalings of the signal waveform
[3,4]. Wideband systems arise in, e.g., underwater acoustic (UWA) systems or wideband terrestrial
radio frequency systems such as ultra wideband (UWB). Due to multipath, a wideband
linear time-varying (LTV) channel can be more accurately described by a multi-scale
multi-lag (MSML) model
[3,8]. Many signaling schemes have been studied for wideband systems. For instance,
[9,10] consider direct-sequence spread spectrum (DSSS). Recently, the use of OFDM for UWA
or UWB has received considerable attention. To counteract the scaling effect due to
Doppler,
[11] proposes a multi-band OFDM system such that within each band, the narrowband assumption
can still be valid. More commonly, many works assume a single-scale multi-lag (SSML)
model for the wideband LTV channel. Based on the SSML assumption, after a resampling
operation the channel can be approximated by a time-invariant channel but subject
to a carrier frequency offset (CFO)
[12,13]. However, since the channel should be more accurately described by an MSML model,
determining the optimal resampling rate is not trivial
[14].

In this article, we consider OFDM transmission based on an MSML model. The resulting
channel, which is a full matrix in the presence of Doppler, will be equalized by means
of the conjugate gradient (CG) algorithm
[15], whose performance is less sensitive to the condition of the channel matrix than,
e.g., a least-squares approach. On the other hand, the convergence rate of CG is inversely
proportional to the channel matrix condition number. This is especially of significance
if a truncated CG is to be used in practice, which halts the algorithm after a limited
number of iterations in order to reduce the overall complexity. Therefore, it is desired
that the channel matrix is well-conditioned to ensure a fast convergence. To this
end, preconditioning techniques can be invoked to enforce the eigenvalues of the channel
matrix to cluster around one
[16]. To achieve a balance between performance and complexity, we restrict the preconditioner
to be a diagonal matrix, whose diagonal entries can be designed by following the steps
given in
[17]. We notice that a circulant preconditioner in the time domain was introduced in
[18], which is equivalent to a diagonal preconditioner in the frequency domain. This preconditioner
is introduced based on a basis expansion model (BEM), which is often used to approximate
the channel’s time-variation for a narrowband system. For a wideband system as considered
in this article, it can be shown that this preconditioner in the frequency domain
is equal to the inverse of the diagonal entries of the frequency-domain channel matrix.

What is not considered in
[17,18] is the resampling operation at the receiver, which is an indispensable and crucial
step for wideband LTV channels. Different from the trivial resampling scheme for SSML
channel models, an optimum resampling method is proposed in
[14] for MSML channels, which aims at minimizing the average error of approximating the
MSML channel by an SSML model. This article studies the resampling from a preconditioning
point of view. It is observed that if the major channel energy is located on the off-diagonals
of the channel matrix, a diagonal preconditioner will deteriorate the channel matrix
condition rather than improve it, thereby reducing the convergence rate of CG instead
of increasing it as opposed to the claim of
[17]. The energy distribution of the channel matrix is governed by the resampling. Different
from
[14], which only considers rescaling the received signal, and
[19], which considers both rescaling and frequency synchronization, this article will
show that for OFDM systems, all these three resampling parameters can have a significant
impact on the system performance (i.e., rescaling, frequency synchronization and time
synchronization). More specifically, we will extend the results of
[19,20] by jointly optimizing these three resampling parameters both in the frequency domain
and the time domain.

Notation: Upper (lower) bold-face letters stand for matrices (vectors); superscripts T, H, and ∗ denote transpose, Hermitian transpose and conjugate transpose, respectively;
we reserve j for the imaginary unit, <k> and ⌈k⌉ for integer rounding and ceiling of a number k, ∥x∥2 for the two norm of the vector x, ∥A∥Fro for the Frobenius norm of the matrix A, [A]k,m for the (k,m)th entry of the matrix A; diag(x) for a diagonal matrix with x on its main diagonal, and ⊙ for the Hadamard product of two matrices.

System model based on an MSML channel

Continuous data model

Suppose that the baseband transmit signal s(t) consists of K subcarriers, and can be written as

(1)

where the data symbol bk is modulated on the kth subcarrier fk =kΔf, for k = 0,1,…,K−1, with Δf being the OFDM subcarrier spacing. With T = 1/(kΔf), KT is the effective duration of an OFDM symbol. The cyclic prefix and postfix are given
as Tpre and Tpost, respectively. The cyclic prefix is assumed to be longer than the delay spread and
the cyclic postfix is long enough to ensure signal completeness in case of scaling,
which will be defined later on. The rectangular pulse u(t) is defined to be 1 within t∈[−Tpre,KT + Tpost] and 0 otherwise. Prior to transmission, s(t) is up-converted to passband, yielding
, where fc denotes the carrier frequency. With sufficient cyclic extensions, the interference
form adjacent OFDM symbols can be neglected and hence we are allowed to consider an
isolated OFDM symbol in this article without loss of generality. Although this article
discusses the scenario when cyclic extensions are used, the analysis can be directly
applied to zero padding OFDM (ZP-OFDM) with minor modifications.

The considered signal is transmitted over a wideband LTV channel, which is assumed
to comprise multiple resolvable paths. The lth path can mathematically be characterized by the following three parameters:
, the path gain; vl, the radial velocity which is uniquely determined by the incident angle of this path;
and τl, the delay due to the propagation time. In compliance with the wideband assumption,
the received signal resulting from the lth path is given by
, where
is the scaling factor with c the speed of the communication medium (normally c ≫ vl) and
is added as a normalization factor. Depending on the sign of vl, the received signal waveform via this path can be either dilated (a negative vl) or compressed (a positive vl).

With a collection of L + 1 paths, the actual received signal
is given by

(2)

where
stands for the passband noise. In the above, if there exist at least two paths l and l’, for which
and/or
, the channel exhibits a multi-scale multi-lag (MSML) character. For a practical channel,
it is realistic to assume that αl∈[1,αmax and τl∈[0,τmaxa, where αmax ≥ 1 and τmax ≥ 0 determines the scale spread and delay spread, respectively. Note that in many
prior works
[12,13], the approximation
for any l≠l′ is adopted for the sake of analytical ease, which gives rise to an SSML model.

The equivalent complex baseband received signal of
is given by

(3)

where w(t) stands for the baseband noise. By substituting (1) in the above, we can rewrite
y(t) as

(4)

where

(5)

which stands for the time-varying channel frequency response seen by the kth subcarrier. From the definition of hk(t), we notice that the kth subcarrier experiences a frequency offset of (αl−1)(fc + fk) over the lth path.

Remark 1

The cyclic prefix is assumed to be longer than the delay spread and the cyclic postfix
has a duration long enough to ensure signal continuity in the observation window for
t∈[0,KT]. Specifically, it is required that u(αlt−αlτl)=1 within this window for all paths. In other words, because u(αlt−αlτl) gives a time support on
, we should then always satisfy
and also
for any l∈{0,1,…,L}. It leads to

(6)

(7)

When the above conditions are satisfied, we are allowed to drop the notation of the
rectangular pulse u(t) embedded in hk(t) in the sequel for the sake of notational ease.

Discrete data model

For MSML channels, discretizing the received signal and achieving time/frequency synchronization
is not trivial
[12,14]. We illustrate such difficulty in Figure
1, where we assume the transmit signal propagates via three paths. Since the received
symbol is the summation of these three paths, it invites the following questions:

1. Which point should we consider as the starting point of the OFDM symbol (time synchronization)?

Figure 1.Illustration of the synchronization and resampling problem; αl stands for the scaling factor due to the lth path, and β for the rescaling factor adopted by the receiver during resampling.

2. What sampling rate should we adopt to discretize the received signal over MSML
channels (rescaling)?

3. What frequency shift should we apply to remove the residual carrier frequency offset
(frequency synchronization)?

These problems can mathematically be described by determining β, ϕ and σ in the following expression

(8)

where β is a positive number within [1,αmax] and βT represents the sampling rate at the receiver; σis the time shift factor, which is used to represent time synchronization; and likewise,
ϕ is the phase shift factor used for frequency synchronization.
is a normalization factor. Later on, we will show that a different choice of (β,ϕ,σ) can influence the energy distribution of the channel matrix significantly. For the
moment, we leave the values of these parameters open to allow for a general treatment
of the problems. It is clear that when (β,ϕ,σ) = (1,0,0), there is no resampling operation carried out.

After resampling, the noiseless sample obtained at the nth time instance in the time domain is given by (see Appendix 2 for the detailed derivation)

(9)

where we use

(10)

to denote the normalized carrier frequency and

(11)

to denote the normalized delay of the lth path; and the discrete channel coefficient is given by

(12)

In (9), the term
corresponds to the residual CFO related with the lth path after resampling; the term
corresponds to the phase changes due to the time shift along the lth path; and the summation
is the adapted version of the transmitted OFDM signal due to the channel time variation
in the lth path.

Let us now stack the received samples
, for n = 0,…,K−1, into a vector
, and similarly let b = [b0,…,bK−1]T. In the noiseless case, it follows that

where the superscript (β,ϕ) in
and (σ) in
reflects the dependence on the specific resampling parameters. This convention will
hold throughout this article.

Interference analysis

Normally speaking, equalization of an OFDM channel is implemented in the frequency
domain. To this end, the received signal
is first transformed into the frequency domain by means of the DFT, which in the
absence of noise yields

(17)

where
stands for the frequency-domain (FD) channel matrix, which is defined as

(18)

with
being its lth component, whose (m, k)th entry is specified as

(19)

where
and
with
.

It is obvious from (16) that in the absence of Dopper effects, i.e., αl = 1 for l = 0,1,…,L, no rescaling and frequency synchronization is necessary, hence β = 1 and ϕ = 0, which leads to a diagonal
with
. In another special case where αl ≡ α for l = 0,1,…,L, we can also enforce a diagonal
by letting β = α and ϕ = 1−α, a scenario considered in, e.g.,
[13]. For a realistic wideband LTV channel, however, the channel energy distribution in
is governed by a Dirichlet kernel, where the center of this Dirichlet kernel is offset
by

(20)

Clearly, such an offset is not only dependent on the Doppler spread α and the carrier frequency fc, but also on the subcarrier frequency fk = kΔf. The dependence of the signal energy offset on the subcarrier index is unique to
wideband channels, and is also referred to as nonuniform Doppler shifts in
[13]. In contrast, the frequency offset for narrowband channels is statistically identical
for all the subcarriers
[2].

The Dirichlet kernel in (16) also suggests that the signal energy is mostly concentrated
in subcarrier
and its nearby subcarriers, and decays fast in subcarriers farther away. To appreciate
how fast the signal energy decays, let us introduce
to quantify the number of subcarriers where most of the energy of bk is located, which can thus be viewed as the bandwidth of
along its kth column.
is obtained as the smallest B for which

(21)

where γ is a positive threshold no larger than 1. In the left plot of Figure
2, the relationship between
and γ for the case β = 1 and α = 0 (no resampling and frequency synchronization) is plotted. It is clear that most
of the signal energy of bk is captured within a limited bandwidth. For example, with a bandwidth
, roughly 98 % of the signal energy of bk is captured. Notably, this bandwidth is almost independent of ξl,F1 and ξl,F2 as suggested by the left plot of Figure
2.

Since each
is roughly banded, it is therefore reasonable to approximate
, which is a weighted sum of different
matrices, also as banded. As an example, we plot in Figure
3 the structure of
, where we assume that there are in total two paths. Obviously, the approximate bandwidth
of
at the kth column, denoted as
, is

which is independent of σ. We refer the reader to Figure
3 for the physical meaning of the notations. It is important to underscore that since
the bandwidth
is dependent on the subcarrier index k, the boundaries of the band are not parallel to each other as in the narrowband case.
A banded approximation of the channel matrix is crucial to many low-complexity equalizers,
e.g.,
[5-7,18]. The equalizer considered in this article will also adopt this approximation to reduce
the complexity. More specifically, we first define a matrix
, whose (mk)th entry is equal to 1 if
, and 0 otherwise, and we then consider the matrix

(23)

as the banded approximation of
.

With the banded approximation, let us rewrite (14) as

(24)

where
.

The above analysis can also be applied in the time domain in an analogous manner.
See Appendix 3 for the details. Here we only want to highlight that, different from
the energy distribution in the FD channel matrix which is influenced by the rescaling
factor β and the phase-shift factor ϕ[c.f. ξl,F1 and ξl,F2in (16)], the energy distribution in the TD channel matrix is affected by the rescaling
factor β and the time-shift factor σ[c.f. ξl,T1 and ξl,T2 in (39)]. However, similarly as the FD channel matrix, we can also understand from
the right subplot of Figure
2 that
is roughly banded along the lth path in the time domain, and so is the overall time-domain channel matrix
.

Channel equalization scheme

Let us now focus on the channel frequency-domain equalization, which is depicted in
Figure
4. In this figure, it is clear that, prior to the equalization, we propose an optimum
resampling operation to achieve
, which is different from
[14,19] as mentioned previously. Specifically, the resampling method proposed in
[14] only considers the rescaling parameter β while
[19] ignores the time-shift parameter σ. Afterwards, the banded matrix
is adopted to approximate
according to the approach mentioned in the last section. Our banded method induces
a non-parallel bandwidth structure which is different from the banded approach used
in narrowband OFDM systems
[5-7,21]. In order to speed up the convergence of the iterative equalization, we then design
a diagonal preconditioner to improve the condition of this banded matrix. It is noteworthy
here that our preconditioner design is adapted from
[17,18] to enhance its suitability for our MSML scenario. Finally, iterative equalization
is proposed on the preconditioned channel matrix. Although we choose the CG method
in this article, other iterative methods can also be applied, such as the LSQR algorithm
[22].

Additionally, we would like to highlight that just as a single-carrier channel can
be equalized in the frequency domain, it is also possible to equalize an OFDM channel
in the time domain. Due to the similarity, we again refer the reader to Appendix 3
for a detailed mathematical derivation of the time-domain method. The question in
which domain the wideband channel should be equalized, shall be addressed in the following
section.

Iterative equalization

To better motivate the other components of our equalization scheme, we first introduce
the channel equalization method itself. A zero-forcing equalizer in the frequency
domain is considered, given by

(25)

where
is the obtained estimate of b. Because the original channel matrix
is a full matrix, its inversion inflicts a complexity of
and is thus not desired for a practical system. To lower the complexity,
has been replaced by the banded approximation
in (22).

Besides, the matrix inversion in (22) will be implemented iteratively using the CG
algorithm. An advantage of using CG rather than inverting the matrix directly is that
the resulting data estimates yielded by CG are always constrained in the Krylov subspace,
making its performance less susceptible to the spectral distribution of
. In practice, a truncated CG, which halts the algorithm after a limited number of
iterations, is desired to further reduce the complexity. It is well-known that the
convergence of the CG algorithm can be accelerated by applying preconditioning on
[16,17,23]. With CFdenoting such a preconditioner, the I/O relationship given in (21) in the noiseless
case can be rewritten as

(26)

from which an estimate of
is first obtained by applying CG on the preconditioned matrix
. Afterwards,
is computed to obtain the final data estimates. For details about our CG equalization,
see Appendix 4.

The optimal design of CFcan be exhaustive
[23]. Inspirited by
[17], we find our preconditioner by minimizing a cost function based on the Frobenius
norm, which clusters most of the eigenvalues of
around 1 with the exception of a few outliers. Further, observing that the design
of CF itself, as well as the operation of
, inflicts an additional complexity, a common approach is to impose a sparse structure
on CF, e.g., diagonal
[17] as

(27)

Diagonal preconditioning

In this section, we will show that the normal approach to design the diagonal preconditioner
as described in
[17] will not necessarily cluster eigenvalues around one. To realize this, let us consider
the diagonal preconditioner CF,⋆ that minimizes the cost function in the Frobenius norm
[17] given by

(28)

which leads to

(29)

where ek is the kth column of the identity matrix.

One problem of the above diagonal preconditioner designed by (25) is that the eigenvalues
may, in some situations, tend to cluster around zero instead of one, with the consequence
that the condition number of the preconditioned channel matrix increases considerably.
To understand this, assume there exists a ε1>0 such that

(30)

for k={0,1,…,K−1}. At the same time, assume there exists a ε0>0 such that

(31)

for k∈{0,…,K−1}.

If we denote the kth eigenvalue of the preconditioned channel matrix
as μk, (27) indicates that (for details see Appendix 5)

(32)

which means that all μk’s lie inside a disk of radius
centered around zero. Similarly, from (26) we have

(33)

which implies that all μk’s at the same time lie inside a disk of
centered around one. It is clear that if ε0<ε1, then minimizing
will at the same time minimize the Frobenius norm
itself, making the eigenvalues more clustered around zero rather than one.

With cF,k,⋆defined in (25), we can show that

(34)

and

(35)

Obviously, if
, for k=0,…,K−1, then the optimal diagonal preconditioner will cluster the eigenvalues in a “wrong”
area. This case arises when the sum of the off-diagonal power in each column is higher
than the power on the diagonal. Such a situation could occur in multi-scale channels
where significant channel power is located on off-diagonal entries as we argued in
the previous section (see Figure
3 for instance). In the upper-left plot of Figure
5, the eigenvalues of such a matrix, with and without preconditioning, are displayed
on a complex plane. It can be seen that diagonal preconditioning indeed clusters the
eigenvalues around zero rather than one.

Figure 5.Left plots: eigenvalues with and without preconditioning; Right plots: convergence
performance with and without preconditioning; FD matrix for top two plots corresponds
to the original channel, FD matrix for bottom two plots is obtained after our optimum
resampling; The MSML channel is set according to Table 1.

To evaluate the impact of such a preconditioner on the convergence of CG, we compute
the mean squared error (MSE) as

(36)

with
being the result obtained at the ith iteration of our CG equalization as mentioned in Appendix 4. In the top-right plot
of Figure
5, it is clear that the CG convergence with such a diagonal preconditioner is even
worse than without any preconditioning. This illustrates that the diagonal preconditioning
defined in (25) may not always yield a better performance than without preconditioning,
as opposed to what is claimed in
[17,18]. Using a more complex structured preconditioner can avoid this, which is, however,
not desired due to complexity and implementation considerations.

To alleviate this problem, we adapt the diagonal preconditioner in (24) and (25) as
follows

(37)

where

(38)

In Section ‘Optimal resampling’, we will show how to enhance (32) with a higher probability
by means of optimal resampling.

Optimal resampling

From the previous subsections, we understand that the effectiveness of a diagonal
preconditioner depends on the energy distribution of the channel matrix. It is desired
that the channel matrix should have most of its energy concentrated on the main diagonal.
The analysis in Section ‘Discrete data model’ learns that the resampling operation
(βϕσ) plays an important role in governing the energy distribution of the channel matrix,
and so far we have left (βϕσ) open for choice. Recall that resampling is a standard step taken in many wideband
LTV communication systems to compensate for the Doppler effect. For example, optimizing
β is considered in
[14], while βand ϕ are jointly optimized in
[21]. In this sense, the optimal resampling proposed in this article can be considered
as a generalization of
[14,21].

Next, we shall discuss how to jointly optimize the resampling parameters (β,ϕ,σ). Focusing on the FD matrix
, we desire
for all k∈{0,1,…,K−1}. However, satisfying the above condition for each index k individually is expensive. As a relaxation, we practically seek

To this end, let us denote the diagonal energy ratio as

(39)

and define our resampling operation by solving

(40)

which leads to the maximal ratio
. One can also explain this resampling as minimizing the total amount of ICI in the
frequency domain.

Since the energy governing mechanism is determined by the sinc function as indicated
in (16), we can equivalently rewrite (34) by only maximizing the diagonal energy of
as

(41)

where again
and
. It is noteworthy that all three parameters, β, ϕ and σ, play a role in (35), indicating that separately considering one or two parameters
as in
[14,21] might lead to a local maximum.

To illustrate our resampling approach in the frequency domain, we consider the channel
example specified in Table
1, where we also compare the properties of the resampled FD channel (i.e., the condition
number and diagonal power ratio of the channel matrix) with the original MSML FD channel.
A geometric interpretation may help to understand our resampling operation since β rotates the FD matrix through
, ϕ shifts the FD matrix through
in (16), and σ influences the phase of each element in (35). The joint effect of these actions maximizes
the matrix diagonal energy. The yielded resampling (βF,⋆ϕF,⋆σF,⋆)=(1.015,−0.015,−15.00) corresponds to a maximal diagonal power ratio
. We underscore that the condition number is already significantly reduced, solely
by the optimum resampling, from 4.26×105to 23.36. In comparison, the resampling method proposed in
[14] yields (βϕσ)=(1.016,0,0) and
. Its corresponding condition number is 432.78, which is larger than our condition
number after resampling. This is not surprising since the criterion adopted in
[14] focuses only on minimizing the aggregate errors between the multi-scale channel and
its single-scale approximation, which is different from our criterion.

In the lower plots of Figure
5, we show the effectiveness of diagonal preconditioning applied to the resampled channel
in Table
1. It is clear that, after our resampling procedure, the diagonal preconditioner clusters
the eigenvalues of the preconditioned FD channel matrix closer to one than without
preconditioning, which further reduces the condition number from 23.36 to 7.17. In
contrast, without optimal resampling, the preconditioner “wrongly” pushes the eigenvalues
closer to zero. In this case, the matrix condition number increases from 4.26×105 to 1.19×106, and hence the CG equalizer performs even worse than without preconditioning as shown
in the top two plots of Figure
5.

Similarly, we can show that optimal resampling can also improve the performance of
the CG in the time domain, for which we just provide Table
2 and Figure
6 here due to space limitations. From them, we can make the same observations as from
Table
1 and Figure
5 for the frequency domain case.

Figure 6.Left plots: eigenvalues with and without preconditioning; Right plots: convergence
performance with and without preconditioning; TD matrix for top two plots corresponds
to the original channel, TD matrix for bottom two plots is obtained after our optimum
resampling; The MSML channel is set according to Table 2.

Frequency-domain or time-domain equalization?

In the previous sections, we showed that the equalization of an OFDM channel can be
implemented in either the frequency or the time domain. With the CG algorithm specified
in Appendix 4, it is clear that the cost of equalization in the frequency domain will
be upper-bounded by
with
for each CG iteration. Likewise, the cost of equalization in the time domain will
be upper-bounded by
with
. By assuming that the number of CG iterations is predetermined and identical in both
domains, we can use the ratio
as a criterion to choose in which domain the equalization will be realized in order
to minimize the complexity.

However, the evaluation of
is cumbersome and lacks the insight of the channel physics. For simplicity reasons,
we equivalently consider the proportion given by

(42)

where we reasonably assume
[see Figure
2. One may argue that the above evaluation is still cumbersome. However, if a realistic
channel allows us to assume, for all l∈{1,2,…,L}, that

(43)

which indicates that the Doppler scale spread is well-limited, it follows that
and
. In other words,
and
, both of which are independent of the symbol index. With these assumptions, ε can further be simplified as

(44)

which suggests that if the maximum difference between the Doppler shifts of each path
(i.e.,
) is smaller than the maximum difference between the time shifts of each path (i.e.,
αlλl), then equalization should be realized in the frequency domain; otherwise, a time-domain
approach will be preferred. A similar conclusion has been made for narrowband systems
[24], though its extension to wideband systems is not straightforward as shown above.

To illustrate the above idea, we again use the channel examples specified in Tables
1 and
2, respectively. We use Brul=5 to roughly capture γ=98% of the channel energy in both domains where γ is introduced in (18). In this way, we have ε≈0.10<1 for the channel in Table
1, while for the channel in Table
2, we have ε≈2.00>1.

For both channels, we compare the equalization performance in different domains. OFDM
with K=128 subcarriers using QPSK is transmitted and the receiver is assumed to have perfect
channel knowledge. We examine the bit error rate (BER) results of our CG equalization
with a fixed CG iteration number (e.g., iF,max=iT,max=100). We use different bandwidths for the banded approximation
and
during the equalization and the values for
and
have also been given in Tables
1 and
2, respectively. After our optimal resampling in either domain, the CG equalization
is carried out using the appropriate preconditioner design.

The left subplot of Figure
7 plots the BER performance as a function of signal-to-noise ratio (SNR) for Channel
I. Note that (βF,⋆,ϕF,⋆,σF,⋆)=(1.015,−0.015,−15) and (βT,⋆,ϕT,⋆,σT,⋆)=(1.015,−0.016,0.00) for this channel. It can be seen that the performance of the
FD equalizer (FDE) based on
outperforms the TD equalizer (TDE) based on
using the same bandwidth
. In other words, FDE is more attractive than TDE in this case.

The BER performance for Channel II is illustrated in the right subplot of Figure
7, where the optimal resampling parameters are (βT,⋆,ϕT,⋆,σT,⋆)=(1.016,−0.021,−1) and (βF,⋆,ϕF,⋆,σF,⋆)=(1.016,−0.016,−3). In this case, it is evident that the TD equalizer is more appealing.

These observations made for the channels in Tables
1 and
2 confirm our metric ε for determining which domain is more suitable for channel equalization. Additionally,
we like to point out that, in either domain, with a larger bandwidth the BER performance
of our CG equalization will be increased.

Numerical results

In this section, we randomly generate two different types of wideband channels as
specified in Table
3: ε<1 (Case I) represents wideband LTV channels where the Doppler differences among the
multipath are more pronounced than the delay differences; and ε>1 (Case II) is the case where the Doppler differences among the multipath are less
pronounced than the delay differences. For all simulations, OFDM with K=128 subcarriers is considered with QPSK. The wideband channels are assumed to have
L=5 paths, whose channel gains (i.e.,
’s) are modeled to be identically and independently distributed. The path delay (τl) is chosen as a random variable that has a uniform distribution within the range
[0,τmax]. Likewise, the path scale (αl) is chosen as a random variable that obeys a uniform distribution within the range
[1,1 + αsp] with αsp the scale spread. For both cases, the receiver is assumed to have perfect channel
knowledge and the cyclic extensions at the transmitter are Tpre=32T and Tpost=10T which satisfy (6) and (7). In all simulations, a banded approximation of the channel
matrix is adopted in both domains with the same bandwidth (e.g.,
).

In Figure
8, the convergence of the CG equalization is plotted in terms of the bit error rate
(BER) against the number of iterations at SNR=30 dB for Case I. Since ε<1, frequency-domain equalization (FDE) is carried out. It is clear that the receiver,
which simply adopts a diagonal preconditioner in (25) without resampling, performs
worst. The performance is already considerably improved if optimal resampling is applied.
Moreover the use of our preconditioner given by (31) boosts the performance even further.

Figure 8.BER versus number of iterations for Case I channels at SNR = 30 dB.

The proposed resampling and preconditioning method can also benefit from other Krylov-based
algorithms. For instance, the LSQR algorithm exploiting a full channel matrix is studied
in
[18]. Note that
[18] focuses on a narrowband LTV system where no resampling is required. Further, the
preconditioner given in
[18] is based on a truncated basis expansion model (BEM) which is usually used for the
approximation of a narrowband time-varying channel. Because it is not clear whether
such a truncated BEM is still suitable for a wideband LTV channel, in order to emulate
a similar approach as in
[18] for constructing the preconditioner, we utilize a (trivial) full-order critically-sampled
complex exponential BEM (the CCE-BEM
[25]) in the simulation. The preconditioner in
[18] then boils down to the inverse of the diagonal of the frequency-domain channel matrix,
which is obviously sub-optimal in the Frobenius norm sense. Consequently, it is no
surprise that directly applying the equalizer of
[18] to wideband LTV channels yields a bad performance as shown in Figure
8. In comparison, the LSQR algorithm benefiting from the optimal resampling and our
preconditioner renders the fastest convergence rate and lowest BER amongst all the
equalization schemes. Of course, such an improved BER performance is achieved by leveraging
the full channel matrix at the cost of a higher complexity, compared to our proposed
method using banded matrices.

Figure
9 exhibits the BER versus SNR for the CG-based equalization schemes, where a truncated
CG is used which halts at the 5th iteration. It can be seen in the figure that the
equalizer leveraging the full channel matrix gives the best BER performance but inflicts
more complexity. When using a banded channel matrix approximation, the frequency-domain
approach performs much better than the time-domain approach because we have ε<1 for this type of channel. Additionally, the equalization approach in
[14] is carried out and its performance is also shown in Figure
9. As we discussed earlier, the resampling operation in
[14] is solely focused on the rescaling parameter ignoring the impact of frequency and
time synchronization, which is therefore sub-optimal. Besides, the equalizer in
[14] approximates the channel matrix to be diagonal (i.e., using a bandwidth of one for
the banded matrices), and thus its performance becomes inferior in the presence of
higher scale differences among the multipath as in the tested channel here.

The performance of the equalizers for Case II is depicted in Figure
10, where the significance of optimal resampling and our adapted preconditioner is again
illustrated just like in Figure
8. Similarly, we can see that the LSQR algorithm in
[18] also works well for this type of channel if optimal resampling and preconditioning
are included.

Different from Case I, the channels of Case II are subject to a larger delay spread
than a Doppler spread (i.e., ε>1). In this case, a time-domain equalizer will be more effective than its frequency-domain
counterpart as validated in Figure
11. The equalizer in
[14] yields a much worse performance than ours since the Doppler scale spread differences
in this case are even higher than for Case I.

Conclusions

In this article, we have discussed iterative equalization of wideband channels using
the conjugate gradient (CG) algorithm for OFDM systems. The channel follows a multi-scale
multi-lag (MSML) model, and suffers therefore from interferences in both the frequency
domain and time domain. To lower the equalization complexity, the channel matrices
are approximated to be banded in both domains. A novel method of optimal resampling
is proposed, which is indispensable for wideband communications. A diagonal preconditioning
technique, that accompanies the CG method to accelerate the convergence, has also
been adapted to enhance its suitability. Experimental results have shown that our
equalization scheme allows for a superior performance to those schemes based on a
single-scale resampling method, without any resampling operation, or using a traditional
preconditioning procedure. In addition, we gave a simple criterion to determine whether
to use a frequency-domain or time-domain equalizer, depending on the channel situation,
to obtain the best BER performance with the same complexity. Such a criterion is also
validated by experiments.

Appendix 1

Detailed derivation of the discrete data model

Here we give the derivation of (9), assuming no noise is present. We start from (8)
given by

(45)

where hk(t) is defined in (5) and the embedded u(t) in hk(t) is considered to be one for the concerned observation window as clarified in Remark
1.

Now, we substitute hk(t) to obtain

(46)

where the channel coefficient is given by

(47)

Now, if we denote

(48)

for the normalized carrier frequency and

(49)

for the normalized delay of the lth path, we have

(50)

which gives (9).

Appendix 2

System model in the time domain and time-domain equalization

To derive the time-domain model, let us rewrite (10) as

(51)

where
, and
stands for the time-domain (TD) channel matrix

(52)

with
being its lth component. The (m,k)th entry of
is given by

(53)

where
and ξl,T2=αl(λl + σ).

Observing the analogy between (16) and (39), a similar interference analysis can be
made on HT. By defining

(54)

we can introduce the symbol
defined as [c.f. (18)]

(55)

which determines the index set of the data symbols that contribute the most to the
mth received signal
via the lth path. Note that
in (18) depends on the resampling factor β and the frequency shift factor ϕ, whereas
in (18) depends on the resampling factor βand the time shift factor σ.

Similarly as in the frequency domain, we obtain a banded approximation of
by introducing

(56)

and a selection matrix
, whose (m,k)th entry is equal to 1 if
, and 0 otherwise. Then the banded approximation of
is obtained by

(57)

We can then rewrite (37) as

(58)

where

The time-domain equalization can be presented in an analogous manner as in the frequency
domain. Similar to its FD counterpart in (23), we here rewrite the noiseless case
for (44) as

(59)

where
, CT is the preconditioner applied in the time domain and
. We first estimate s by applying the CG algorithm on
to invert
iteratively, and afterwards we obtain
.

We highlight that the adopted diagonal preconditioner CT,⋆=diag{[cT,0,⋆,cT,1,⋆,…,cT,K−1,⋆]T} is defined in a similar manner as in the frequency domain. Specifically, we use

(60)

where

(61)

To enhance the suitability of the preconditioner, the optimal resampling operation
is needed as given by

(62)

Appendix 3

Equalization using the conjugate gradient algorithm

If we consider to solve the preconditioned system in (23) in a similar manner as (22),
we have

(63)

where
, and
is the estimate of
.

Its implementation using CG is described in the frequency domain as follows

1. Define
and i=0;

2. Perform the following iterations:

(64)

where a(0)=g(0)=dF,
and
;

3. Perform
, which is the ith output of the equalization process, and the index i is incremental from 0 to imax where imax is the iteration number when the stopping criterion of the CG is satisfied.

Notably, the optimal stopping criterion for CG can be case dependent, e.g., as discussed
in
[23], and is not included in this article. When our CG iterations stop, we finally have
, which is the data estimate.

It is worthy to note that the computational complexity of each CG iteration above
is determined by the complex multiplication (CM) of
with a vector (e.g.
or a(i)), e.g., as in (49). When
is a diagonal preconditioner as considered in this article, the bandwidth of the
preconditioned
equals that of
, and consequently
is banded with a bandwidth
where
with
defined in (19). In this case, the computational complexity of each iteration is
upper-bounded by
which is linear in the vector size K.

One can also repeat the above derivations using the TD notations for the TD CG equalization.

Appendix 4

Eigenvalue locations

We consider the diagonal matrix CF=diag{[cF,0,cF,1,…,cF,K−1]T}, and denote the eigenvalues of
as {μ1,μ2,…,μK−1}.

Let UWU be a Schur decomposition of
such that UUH=IK×K and the diagonal elements of W equal {μ1,μ2,…,μK−1}. Then

(65)

Note that
, where ekstands for an all-zero vector except for its kth entry which equals 1, as defined in (25) for k=0,1,…,K. We then recall (27), which holds for any k∈{0,1,…,K−1}, and thus upper-bounds the above expressions as

(66)

Similarly, we can also prove that
associated with (26).

Endnote

a As a matter of fact, the case where αl<1 or τl<0 can be converted to the current situation by means of proper resampling and timing
at the receiver. This justifies the assumption of a compressive and causal scenario
without loss of generality.

Competing interests

The authors declare that they have no competing interests.

Acknowledgements

The first author wants to thank the National University of Defense Technology, China,
and also the China Scholarship Council for the financial support. This work was supported
in part by NWO-STW under the VICI program (project 10382). The work of Z. Tang is
also supported in part by the European Defence Agency (EDA) project RACUN (Robust
Acoustic Communication in Underwater Networks). In addition, we would like to thank
Dr. Magnus Lundberg Nordenvaad from the Lulea University of Technology, Sweden, Prof.
Urbashi Mitra from the University of Southern California, U.S., and Prof. Huihuang
Chen from the Xiamen University, China, who participated in valuable discussions.